11/11/2015Physics 201, UW-Madison1 Physics 201: Chapter 14 – Oscillations (cont’d) General...

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Transcript of 11/11/2015Physics 201, UW-Madison1 Physics 201: Chapter 14 – Oscillations (cont’d) General...

  • *Physics 201, UW-Madison*Physics 201: Chapter 14 Oscillations (contd)

    General Physical Pendulum & Other ApplicationsDamped OscillationsResonances

    Physics 201, UW-Madison

  • *Physics 201, UW-Madison*Simple Harmonic Motion: Summary Solution: = A cos(t + )

    Force:kms0

    Physics 201, UW-Madison

  • Question 1*Physics 201, UW-Madison*The amplitude of a system moving with simple harmonic motion is doubled. The total energy will then be4 times larger2 times largerthe same as it washalf as muchquarter as much

    Physics 201, UW-Madison

  • Question 2*Physics 201, Fall 2006, UW-Madison*A glider of mass m is attached to springs on both ends, which are attached to the ends of a frictionless track. The glider moved by 0.2 m to the right, and let go to oscillate. If m = 2 kg, and spring constants are k1 = 800 N/m and k2 = 500 N/m, the frequency of oscillation (in Hz) is approximately6 Hz2 Hz4 Hz8 Hz10 Hz

    Physics 201, Fall 2006, UW-Madison

  • *Physics 201, UW-Madison*The harmonic oscillator is often a very good approximation for (non harmonic) oscillations with small amplitude:

    Physics 201, UW-Madison

  • General Physical Pendulum*Physics 201, UW-Madison*Suppose we have some arbitrarily shaped solid of mass M hung on a fixed axis, and that we know where the CM is located and what the moment of inertia I about the axis is.The torque about the rotation (z) axis for small is (using sin ) = -Mg d -Mg R

    dMgz-axisRxCMwhere

    Physics 201, UW-Madison

  • Torsional Oscillator*Physics 201, UW-Madison*Consider an object suspended by a wire attached at its CM. The wire defines the rotation axis, and the moment of inertia I about this axis is known. The wire acts like a rotational spring.When the object is rotated, the wire is twisted. This produces a torque that opposes the rotation.In analogy with a spring, the torque produced is proportional to the displacement: = -k

    Physics 201, UW-Madison

  • Torsional Oscillator*Physics 201, UW-Madison*Since = -k, = I becomeswhereThis is similar to the mass on spring Except I has taken the place of m (no surprise).

    Physics 201, UW-Madison

  • *Physics 201, UW-Madison*Damped OscillationsIn many real systems, nonconservative forces are presentThe system is no longer idealFriction/drag force are common nonconservative forcesIn this case, the mechanical energy of the system diminishes in time, the motion is said to be damped

    The amplitude decreases with timeThe blue dashed lines on the graph represent the envelope of the motion

    Physics 201, UW-Madison

  • *Physics 201, UW-Madison*Damped Oscillation, ExampleOne example of damped motion occurs when an object is attached to a spring and submerged in a viscous liquidThe retarding force can be expressed as F = - b v where b is a constantb is called the damping coefficient The restoring force is kxFrom Newtons Second LawFx = -k x bvx = maxor,

    Physics 201, UW-Madison

  • *Physics 201, UW-Madison*Damped Oscillation, Example

    When b is small enough, the solution to this equation is:

    with

    (This solution applies when b

  • Energy of weakly damped harmonic oscillator:The energy of a weakly damped harmonic oscillator decays exponentially in time.*Physics 201, UW-Madison*The potential energy is kx2:Time constant is the time for E to drop to 1/e .

    Physics 201, UW-Madison

  • *Physics 201, UW-Madison*Types of Damping0 is also called the natural frequency of the systemIf Fmax = bvmax < kA, the system is said to be underdampedWhen b reaches a critical value bc such that bc / 2 m = 0 , the system will not oscillateThe system is said to be critically dampedIf Fmax = bvmax > kA and b/2m > 0, the system is said to be overdampedFor critically damped and overdamped there is no angular frequency

    Physics 201, UW-Madison

  • *Physics 201, UW-Madison*Forced OscillationsIt is possible to compensate for the loss of energy in a damped system by applying an external sinusoidal force (with angular frequency )The amplitude of the motion remains constant if the energy input per cycle exactly equals the decrease in mechanical energy in each cycle that results from resistive forces

    After a driving force on an initially stationary object begins to act, the amplitude of the oscillation will increaseAfter a sufficiently long period of time, Edriving = Elost to internal Then a steady-state condition is reached The oscillations will proceed with constant amplitude

    The amplitude of a driven oscillation is

    0 is the natural frequency of the undamped oscillator

    Physics 201, UW-Madison

  • *Physics 201, UW-Madison*ResonanceWhen 0 an increase in amplitude occursThis dramatic increase in the amplitude is called resonanceThe natural frequency w0 is also called the resonance frequencyAt resonance, the applied force is in phase with the velocity and the power transferred to the oscillator is a maximumThe applied force and v are both proportional to sin ( t + )The power delivered is F . vThis is a maximum when F and v are in phaseResonance (maximum peak) occurs when driving frequency equals the natural frequencyThe amplitude increases with decreased dampingThe curve broadens as the damping increases

    Physics 201, UW-Madison

  • Resonance Applications:*Physics 201, Fall 2006, UW-Madison*Extended objects have more than one resonance frequency. When plucked, a guitar string transmits its energy to the body of the guitar. The bodys oscillations, coupled to those of the air mass it encloses, produce the resonance patterns shown.

    Physics 201, Fall 2006, UW-Madison

  • Question 3*Physics 201, Fall 2006, UW-Madison*A 810-g block oscillates on the end of a spring whose force constant is k=60 N/m. The mass moves in a fluid which offers a resistive force proportional to its speed the proportionality constant is b=0.162 N.s/m.

    Write the equation of motion and solution.

    What is the period of motion?

    Write the displacement as a function of time if at t = 0, x = 0 and at t = 1 s, x = 0.120 m.0.730 s

    Physics 201, Fall 2006, UW-Madison

    frequency = omega/(2 pi) = sqrt(1300/2)/(2 pi)=4 Hz